Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,23)(17,22)(18,21)(19,26)(20,25)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15,17,26)(14,19,22,24)(16,27,18,25)(20,21,28,23);(7,12)(9,11)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,24)(22,28);(6,10)(9,11)(13,16,17,18)(14,21,22,23)(15,25,26,27)(19,20,24,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482606);